If Josiah made the trip in 2 hours less than Davin, how far apart do Josiah and Davin live?

Question

Davin and Josiah decided to meet up, but Davin thought they were meeting in Josiah's hometown and Josiah thought they were meeting in Davin's hometown! They each left and made the same trip in opposite directions at 9am. Davin drove at 50 miles (mi) per hour and Josiah drove 10 mph faster. If Josiah made the trip in 2 hours less than Davin, how far apart do Josiah and Davin live?

Answer 1

Use the formula: Rate * Time = Distance

Let x equal Davin's time in hours. Then x - 2 equals Josiah's time because Josiah arrived two hours earlier.

Davin's rate is 50 mph. Therefore, Josiah's rate is 60 mph because Josiah drove 10 mph faster.

Using the above formula, Davin's expression is 50 * (x).

Using the same formula, Josiah's expression is 60 * (x - 2).

Because the distance between both hometowns is the same, Davin's expression and Josiah's expression can be set equal to one another:

50 * (x) = 60 * (x - 2)

Simplify the expression on either side of the equation:

50x = 60x - 120

Subtract the quantity 60x from both sides of the equation:

50x - 60x = 60x - 60x -120

-10x = -120

Divide both sides of the equation by -10:

-10x/-10 = -120/10

x = 12

Because x = 12, the distance between the hometowns can be found by multiplying 12 by 50:

50 * 12 = 600

Therefore, Davin and Josiah live 600 miles apart.

Answer 2

If Josiah is traveling 10 mph faster then his speed is 60 mph. It says they made the same trip, which is another way of saying they travelled the same distance, let's call this distance d. We should know that speed x time = distance, so for Davin:

50t = d

And for Josiah:

60(t-2) = d

Note that since Josiah completed the trip two hours faster, his total time travelled is t-2 if we're saying Davin's travel time is t. The problem only asks for the distance travelled, not the time travelled, so we do not need to solve for t. We can substitute it out by noting from Davin's equation that:

t = d/50

We substitute this into Josiah's equation to get:

60(d/50 - 2) = d

6/5*d -120 = d

6/5*d - d = 120

1/5*d = 120

d = 5 *120

d = 600

So we see that the total distance travelled was 600 miles. We can check our answer by now solving for the times travelled. For Davin:

t = 600/50

t = 12 h

For Josiah:

t = 600/60

t = 10 h

We see that Josiah does complete the 600 mile trip two hours faster than Davin and so we can be confident our answer is correct.